Last syzygies of 1-generic spaces

We consider determinantal varieties X(γ) of expected codimension defined by the maximal minors of a matrix M(γ) of linear forms representing a linear map γ. Eisenbud and Popescu have conjectured that 1-generic linear maps γ have the property that the syzygy ideals I(s) of all last syzygies s of X(γ) coincide with IX(γ). We prove a geometric version of this conjecture: for 1-generic linear maps γ the syzygy varieties Syz(s)=V(I(s)) of all last syzygies have the same support as X(γ).